Functional integration: Difference between revisions

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}}</ref> and [[Norbert Wiener]] in a series of studies culminating in his articles of 1921 on [[Brownian motion]]. They developed a rigorous method (now known as the [[Wiener measure]]) for assigning a probability to a particle's random path. [[Richard Feynman]] developed another functional integral, the [[path integral formulation|path integral]], useful for computing the quantum properties of systems. In Feynman's path integral, the classical notion of a unique trajectory for a particle is replaced by an infinite sum of classical paths, each weighted differently according to its classical properties.

Functional integration is central to quantization techniques in theoretical physics. The algebraic properties of functional integrals are used to develop series used to calculate properties in [[quantum electrodynamics]] and the [[standardStandard modelModel]] of particle physics.

==Functional integration==

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Whereas standard [[Riemann integral|Riemann integration]] sums a function ''f''(''x'') over a continuous range of values of ''x'', functional integration sums a [[functional (mathematics)|functional]] ''G''[''f''], which can be thought of as a "function of a function" over a continuous range (or space) of functions ''f''. Most functional integrals cannot be evaluated exactly but must be evaluated using [[perturbation methods]]. The formal definition of a functional integral is

<math display="block">

\int G[f]\; \mathcal{D}[f] \equiv \int_{-\inftymathbb{R}}^\infty \cdots \int_{-\inftymathbb{R}}^\infty G[f] \prod_x df(x)\;.

</math>

However, in most cases the functions ''f''(''x'') can be written in terms of an infinite series of [[orthogonal functions]] such as <math>f(x) = f_n H_n(x)</math>, and then the definition becomes

<math display="block">

\int G[f] \; \mathcal{D}[f] \equiv \int_{-\inftymathbb{R}}^\infty \cdots \int_{-\inftymathbb{R}^\infty} G(f_1,; f_2,; \ldots) \prod_n df_n\;,

</math>

which is slightly more understandable. The integral is shown to be a functional integral with a capital ''<math>\mathcal{D''}</math>. Sometimes itthe argument is written in square brackets: [''Df''] or ''<math>\mathcal{D''}[''f'']</math>, to indicate thatthe ''f''functional isdependence aof the function in the functional integration measure.

==Examples==

Most functional integrals are actually infinite, but thenoften the limit of the [[quotient]] of two related functional integrals can still be finite. The functional integrals that can be evaluated exactly usually start with the following [[Gaussian integral]]:

:<math>

\frac{\displaystyle\int \exp\left\lbrace-\frac{1}{2} \intint_{\mathbb{R}}\left[\int_{\mathbb{R}} f(x) K(x,;y) f(y) \,dx\,dy + \int J(x) f(x) \,right]dx\right\rbrace \mathcal{D}[f]}

{\displaystyle\int \exp\left\lbrace-\frac{1}{2} \intint_{\mathbb{R}^2} f(x) K(x,;y) f(y) \,dx\,dy\right\rbrace \mathcal{D}[f]} =

\exp\left\lbrace\frac{1}{2}\intint_{\mathbb{R}^2} J(x) \cdot K^{-1}(x,;y) \cdot J(y) \,dx\,dy\right\rbrace.\,,

</math>

in which <math>

By functionally differentiating this with respect to ''J''(''x'') and then setting to 0 this becomes an exponential multiplied by a polynomial in ''f''. For example, setting <math>K(x, y) = \Box\delta^4(x - y)</math>, we find:

K(x;y)=K(y;x)

</math>. By functionally differentiating this with respect to ''J''(''x'') and then setting to 0 this becomes an exponential multiplied by a polynomialmonomial in ''f''. ForTo example,see setting <math>K(xthis, y)let's =use \Box\delta^4(x - y)</math>,the wefollowing findnotation:

<math>

G[f,J]=-\frac{1}{2} \int_{\mathbb{R}}\left[\int_{\mathbb{R}} f(x) K(x;y) f(y)\,dy + J(x) f(x)\right]dx\, \quad,\quad W[J]=\int \exp\lbrace G[f,J]\rbrace\mathcal{D}[f]\;.

</math>

With this notation the first equation can be written as:

<math>

\dfrac{W[J]}{W[0]}=\exp\left\lbrace\frac{1}{2}\int_{\mathbb{R}^2} J(x) K^{-1}(x;y) J(y) \,dx\,dy\right\rbrace.

</math>

Now, taking functional derivatives to the definition of <math>

W[J]

</math> and then evaluating in <math>

J=0

</math>, one obtains:

<math>

\dfrac{\delta }{\delta J(a)}W[J]\Bigg|_{J=0}=\int f(a)\exp\lbrace G[f,0]\rbrace\mathcal{D}[f]\;,

</math>

<math>

\dfrac{\delta^2 W[J]}{\delta J(a)\delta J(b)}\Bigg|_{J=0}=\int f(a)f(b)\exp\lbrace G[f,0]\rbrace\mathcal{D}[f]\;,

</math>

<math>

\qquad\qquad\qquad\qquad\vdots

</math>

which is the result anticipated. More over, by using the first equation one arrives to the useful result:

:<math>

\dfrac{\displaystyle\int f(a) f(b) edelta^2}{i \intdelta fJ(xa) \Boxdelta fJ(xb) \,d^4x} \mathcalleft(\dfrac{DW[J]}{W[f0]}\right)\Bigg|_{J=0}=

K^{-1}(a; b)\;;

{\displaystyle\int e^{i \int f(x) \Box f(x) \,d^4x} \mathcal{D}[f]} =

</math>

K^{-1}(a, b) = \frac{1}{|a - b|^2},

Putting these results together and backing to the original notation we have:

<math>

\frac{\displaystyle\int f(a)f(b)\exp\left\lbrace-\frac{1}{2} \int_{\mathbb{R}^2} f(x) K(x;y) f(y)\, dx\,dy\right\rbrace \mathcal{D}[f]}

{\displaystyle\int e^\exp\left\lbrace-\frac{i1}{2} \intint_{\mathbb{R}^2} f(x) \Box fK(x;y) f(y) \,d^4x}dx\,dy\right\rbrace \mathcal{D}[f]} =

K^{-1}(a;b)\,.

</math>

where ''a'', ''b'' and ''x'' are 4-dimensional vectors. This comes from the formula for the propagation of a photon in quantum electrodynamics. Another useful integral is the functional [[delta function]]:

:<math>

\int \exp\left\lbrace \intint_{\mathbb{R}} f(x) g(x)dx\right\rbrace \mathcal{D}[f] = \delta[g] = \prod_x\delta\big(g(x)\big),

</math>

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*[[Victor Popov]], Functional Integrals in Quantum Field Theory and Statistical Physics, Springer 1983

*[[Sergio Albeverio]], Sonia Mazzucchi, A unified approach to infinite-dimensional integration, Reviews in Mathematical Physics, 28, 1650005 (2016)

*[[John R. Klauder|Klauder, John]]. "[https://www.phys.ufl.edu/functional-integration/ Lectures on Functional Integration]." ''University of Florida.'' [https://web.archive.org/web/20240708182058/http://www.phys.ufl.edu/functional-integration/ Archived] on July 8th, 2024.